ON THE MOTION OF PENDULUMS. 39
By virtue of (67), udy - vd% is an exact differential. Let tlien udy — vdx = dx ......................... (68).
Eliminating p by differentiation from the two equations (66), and expressing u and v in terms of % in the resulting equation, we get
d*d* ld\d*
"0 ............ (69)'
and, as before
% = %i + %2 .......................... (70),
where
<71>-
We get from (66) and (68)
d I f d* 1 d
d (ff d* 1 d •dx^^-^
which becomes by means of (70), (71), and (72)
^
dx y dt dy
26. Passing to polar co-ordinates r, 6, where d is supposed to be measured from the axis of x, we get from (68), (71), (72), and (73)
JBrdd-®dr = dx ........................ (74),
Id 1 + f^ + ?
Id Id" 1 d
dt \ dr
..................
^ J
R, @ in (74) being the velocities along and perpendicular to the radius vector.